Theorem sbcieg 3781

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2180, ax-12 2216. (Revised by GG, 12-Oct-2024.)

Hypothesis

Ref	Expression
sbcieg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcieg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem sbcieg

Step	Hyp	Ref	Expression
1		df-sbc 3743	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		sbcieg.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	elabg 3633	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
4	1, 3	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  {cab 2741  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-sbc 3743

This theorem is used by:  sbcie  3783  2nreu  4405  reuprg0  4663  rabsnif  4684  ralrnmptw  7121  ralrnmpt  7123  fpwwe2lem3  10748  nn1suc  12387  opfi1uzind  14686  mndind  19056  fgcl  24230  cfinfil  24245  csdfil  24246  supfil  24247  fin1aufil  24284  ifeqeqx  33213  nn0min  33485  bnj1452  35765  cdlemk35s  42197  cdlemk39s  42199  cdlemk42  42201  2nn0ind  44156  zindbi  44157  prproropreud  48806